First off...
There can only be one focal point for a lens. An "infinte focus" is possible, but in the case of a projector, this is more or less unacceptable. The smallest detail that it then becomes possible to see is the aperature of the lens.
Take your fresnel outside and focus an image of the Sun on the ground. Melt some snow or something.
now move the fresnel towards or away from the ground. You don't get a larger image of the Sun, you get a blurry light. This will be the same result from your projector without a triplet.
In order to even give this a fighting shot, you'd need to place your 317mm focal length fresnel ~317mm or a bit further from the LCD, in effect using it as a projection lens. In this configuration, the
absolute best that you could hope for is the same result as using a singlet lens as a projection lens, and that's only if the fresnel is kept optically flat, and it's an extremely well made fresnel. You would also need to be able to adjust that distance to get a projection at any distance.
V&J's earlier example of a pinpoint at 317mm, original LCD size at 634mm, etc is incorrect. There is no distance at which the LCD will properly focus to a pinpoint, with the fresnel anywhere near the LCD. The fresnel would need to be at a distance greater than it's own focal length from the LCD in order to focus at all, let alone to a pinpoint. For what it's worth, doing your melting snow (or frying stuff, if you're somewhere warm enough not to have snow, now) with sunlight doens't focus to a pinpoint, either. It focuses to a small image of the Sun at a magnification proportional to the focal length of the lens and the distance from that lens to the light source (In this case, the Sun.) There is an important difference...
Focusing happens when the SCATTERED rays from a point source are collected over the surface of the lens and redirected to a single point, and is a separate instance for each and every single point source withint your picture. In the case of a projector, each of those is considered to be an individual pixel. Only at the distance where the light from each pixel focuses apart from the light from a different pixel will an image be focused, and this is a very specific distance for a given lens. The
only thing that can "focus" at any distance is a pinhole camera, and this is only the case because there is no lens effect.
Ideally, the field fresnel should have NOTHING to do with the projection lens optics, and in the case where the fresnels are together behind the LCD, this is more or less the case. The field fresnel's chief purpose in life is to collect as much light as possible and direct it towards the projection lens, acting as a light amplifier and not as a focusing lens. There are only 2 reasons for putting the field fresnel in front of the LCD. 1. To keep the light going through the LCD as straight as possible, and 2. To make use of the optical interaction for keystoning adjustment. It actually impedes the focusing of the triplet and results in (slightly) poorer image quality, which is acceptable because of the other benefits.
There is an optical interaction, which modifies the effctive focal length of the projection lens, proportional to the distance from the LCD. This is the case when any two lenses interact, such as when a person wears glasses. We can (and do) use this effect in a projector for keystoning, where we use the magnification effect of the lens to make a part of the screen seem larger to the projection lens. This is only possible within a fraction of the lens' focal elngth.
Well, enough on the essay for now... I'm sure half of the people looking at this stopped reading some time ago.